Market risk of BRIC Eurobonds in the financial crisis period • Dimitrios I. Vortelinos , , • Geeta Lakshmi 1, • Lincoln Business School, University of Lincoln, UK Received 2 March 2014, Revised 23 April 2015, Accepted 23 April 2015, Available online 5 May 2015 doi:10.1016/j.iref.2015.04.012 Highlights • Most significant in terms of risk and jumps are the Chinese, among BRIC Eurobonds. • Most significant range estimator is the Yang Zhang estimator. • Higher risk and jumps for theoretical and not actual prices • Higher expiry period relates to more significant risk and jumps. Abstract The market risk of returns for BRIC Eurobonds has not been thoroughly analyzed via nonparametric estimation methods. The significance of risk and jumps is examined in a monthly sampling frequency. A detailed comparison upon significance of risk and jumps between BRIC Eurobonds is provided. Comparison concerns risk and jumps during the international financial crisis period: February 2007 up to February 2010. Among the BRIC countries, Chinese Eurobonds are the most significant in terms of both risk and jumps. The most significant estimator is the monthly Yang & Zhang range across the set of BRIC Eurobonds. The shorter the expiry period, the higher is the significance of risk and jumps. This is evident in all BRIC Eurobonds. Risk and jump estimates are higher for theoretical prices rather than for actual prices according to all risk and jump significance measures. Keywords
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Market risk of BRIC Eurobonds in the financial crisis period • Dimitrios I. Vortelinos, ,
• Geeta Lakshmi1, • Lincoln Business School, University of Lincoln, UK
Received 2 March 2014, Revised 23 April 2015, Accepted 23 April 2015, Available online 5 May 2015
doi:10.1016/j.iref.2015.04.012
Highlights
•
Most significant in terms of risk and jumps are the Chinese, among BRIC Eurobonds.
•
Most significant range estimator is the Yang Zhang estimator.
•
Higher risk and jumps for theoretical and not actual prices
•
Higher expiry period relates to more significant risk and jumps.
Abstract
The market risk of returns for BRIC Eurobonds has not been thoroughly analyzed via
nonparametric estimation methods. The significance of risk and jumps is examined in a
monthly sampling frequency. A detailed comparison upon significance of risk and jumps
between BRIC Eurobonds is provided. Comparison concerns risk and jumps during the
international financial crisis period: February 2007 up to February 2010. Among the BRIC
countries, Chinese Eurobonds are the most significant in terms of both risk and jumps. The
most significant estimator is the monthly Yang & Zhang range across the set of BRIC
Eurobonds. The shorter the expiry period, the higher is the significance of risk and jumps.
This is evident in all BRIC Eurobonds. Risk and jump estimates are higher for theoretical
prices rather than for actual prices according to all risk and jump significance measures.
Keywords
• BRIC Eurobonds;
• Risk;
• Jumps;
• Bond pricing;
• Financial crisis
JEL classification
• G12;
• G15;
• G17
1. Introduction
In the last decade, the BRIC2 countries have been researched extensively in the economics
and finance literature. One of the first studies researching the significant role of the BRIC
economies in the contemporary international economy's structure is Julius (2005).
Recently, Aloui, Aissa, and Nguyen (2011) showed strong evidence of time-varying
dependence between each of the BRIC markets and the US markets. Some of more recent
studies on BRIC countries are: Cakir and Kadundi (2013) and Bekiros (2014). Fang and You
(2014) investigated how explicit structural shocks that characterize the endogenous character
of changes in oil prices affect three of the four BRICs' stock-market returns. Part of this BRIC
literature is the BRIC Eurobonds3 literature, which has not been extensively investigated. A
recent paper studying the BRIC countries' debt markets is by Steinbock (2012). Speicifically
in ths paper, the prospects for BRIC countries from the Eurozone debt crisis are
studied. Peristiani and Santos (2010) reported that the extent of the dominance of the US
Eurobond market globally has been reduced as the role of BRIC countries in the international
Eurobond market increased. In this paper, BRIC Eurobonds are analyzed using both actual
market prices and theoretical prices. Actual prices are the ones obtained in the market.
Theoretical prices are obtained by a pricing model (as suggested by McCulloch, 1971) which
involves fitting a smooth discount function (which is a cubic spline). Moreover, literature has
also not extensively examined the market risk of BRIC Eurobonds. The present paper
examines the significance of both market risk and jumps of risk series in the recent financial
crisis period4.
Market risk is measured by conditional variance (volatility) that is latent; so, market risk is not
directly observable. Literature has concentrated on parametric estimators of volatility, like: (i)
Generalized AutoRegressive Conditional Heteroskedasticity (GARCH), (ii) Stochastic
Volatility (SV), (iii) Exponentially Weighted Moving Average (EWMA) models, among others.
The ex post volatility essentially becomes observable, if the effect of the microstructure noise
is low. Contemporary realized volatility estimators, as the ones employed here, minimize such
effect. As volatility becomes observable, it can be modeled directly. Andersen and Bollerslev
(1998) introduced the first and most naive realized volatility estimator, as the best
nonparametric volatility estimator. Recent literature suggests that the realized volatility
estimator is useful for: (i) predicting future volatility (Byun & Kim, 2013); (ii) asset allocation in
portfolios (Bandi, Russell, & Zhu, 2008); (iii) risk management (Giot & Laurent, 2004); and (iv)
VaR computation (Clements, Galvao, & Kim, 2008). Sevi (2015) investigated the realized
volatility usefulness for modeling the convenience yield. Specifically, monthly realized
volatilities and jumps explained convenience yield; whereas, jumps were detected as
in Tauchen and Zhou (2010).
The present paper employs the estimation strength of many non-parametric volatility
estimators; all belonging to the realized volatility literature. Estimators are classified into three
groups: realized volatility estimators, range volatility estimators, and realized range-based
volatility estimators. The first group of estimators studied is realized volatility estimators. The
non-parametric estimator that most effectively uses data for estimation purposes is realized
volatility. Andersen et al., 2001 and Andersen et al., 2003 were the first to theoretically and
empirically research realized volatility estimation. There are different parameterizations for the
realized volatility estimation literature. Most of the finance literature estimates realized
volatility in a daily frequency via intraday data series. However, the present paper estimates
monthly volatilities via daily data, because of low intraday and daily liquidity. Jiang and Tian
(2005) out-of-sample compares the realized volatility to implied volatility in a monthly
frequency. A recent influential study in monthly realized volatility estimation (as well as
forecasting) is Busch, Christensen, and Nielsen (2011). A more recent and applied study on
realized volatility estimators is Bollerslev, Osterrieder, Sizova, and Tauchen (2013). Another
nonparametric volatility estimator is range. The first range estimator was suggested
in Parkinson (1980). A recent study in range estimators of volatility is Louzis, Xanthopoulos-
Sisinis, and Refenes (2013). A third group of nonparametric estimators is realized range-
based volatility estimators. One of the very first papers to research this type of estimators
is Martens and van Dijk (2007). A recent study in realized range-based estimators
is Bannouh, Martens, and van Dijk (2013).
In the present paper, twelve nonparametric volatility estimators estimate risk. These
estimators are split into three categories: realized volatility, monthly range, and realized
range-based volatility. The first group includes the 5-minute unrestricted realized volatility
(RVt(m)), the realized bipower variation (BPVt(m)), a moving average-based volatility that uses
the first order residuals (RVt(ma. adj1)), and a moving average-based volatility that uses the
second order residuals (RVt(ma. adj2)). The monthly ranges group includes the monthly
Parkinson range (MRt(Par)), the monthly Garman & Klass range (MRt(GK)), the monthly Rogers
& Satchell range (MRt(RS)), and the monthly Yang & Zhang range (MRt(YZ)). The third group of
realized range-based volatility includes the realized Parkinson range-based volatility (RRt(Par)),
the realized Garman & Klass range-based volatility (RRt(GK)), the realized Rogers & Satchell
range-based volatility (RRt(RS)), and the realized Yang & Zhang range-based volatility
(RRt(YZ)).
Risk (volatility) series is not a continuous process; jumps make this process discontinuous.
Jumps can be detected in any time interval. Literature detected and studied jumps in volatility
from an intraday sampling frequency5 up to a monthly frequency6. The present paper studies
monthly jumps (in monthly volatility series). The employed detection scheme was introduced
in Ait-Sahalia and Jacod (2009)7.
The present paper estimates volatility in a monthly frequency, because most of market
participants in the Eurobonds market do not aim at intraday capital gains. They mostly trade
sovereign bonds either in a daily or most probably in a monthly frequency. Most of them are
fund managers or treasurers, pension or hedge fund managers or treasurers, rebalancing
their portfolios monthly. So, there is no need to employ intraday high-frequency data.
Moreover, there is low liquidity of Eurobonds in an intraday frequency. In this paper, we use
non-parametric volatility estimators, as literature suggests due to higher robustness. Studies
in the literature have recently estimated realized volatility in a monthly frequency. Afonso,
Gomes, and Taamouti (2014) used, as an alternative to parametric volatility models, non-
parametric measures of volatility: the absolute value and the squared returns as proxies of
monthly volatilities. An, Ang, Bali, and Cakici (2014)employed the monthly realized volatility
estimates as a factor in the cross-sectional relation between implied volatility shocks. Zhu and
Lian (2015) provided in a monthly frequency two analytical closed-form formulae for the price
of forward-start variance swap with the realized variance being defined by the actual-return
realized variance and the log-return realized variance. Moreover, Lee, Paek, Ha, and Ko
(2015)employed a structural VAR model for examining the relations among monthly realized
volatility, market return, and aggregate equity fund flows in an international context. Seo and
Kim (2015) examined the effect of investor sentiment on the relationship between the option-
implied information and the future stock return monthly realized volatility. Moreover, more
accurate estimators are employed for nonparametricaly estimating monthly realized volatility
in this paper.
This study contributes to the literature through the following aspects. To the best of our
knowledge, this present study is the first to nonparametrically examine the significance of risk
and jumps of BRIC Eurobonds. Secondly, many realized volatility estimators are employed.
Risk is estimated via twelve nonparametric estimators as split into three groups (realized
volatility, range, and realized range-based volatility). The significance of risk is measured via
the mean magnitude of risk ( ) and the mean Sharpe ratio ( ) as well. Thirdly, two jump
detecton schemes are employed for risk jumps. The significance of jumps is measured via the
mean magnitude of jumps ( ), the mean magnitude of the jump component of risk relative to
the magnitude of the continuous component ( ), the average frequency of jump occurrence (
), and the average frequency of occurrence of statistically significant jumps ( ). Fourthly,
volatility estimates concern both actual and theoretical prices.
The remainder of the article is structured as follows. The second section describes the data
and provides a descriptive analysis of returns. The third section presents the methodology.
The fourth section discusses on empirical findings. The fifth section summarizes and
concludes.
2. Data
2.1. Data description
The sample covers the period from February 2007 to February 2010, a total of 717 trading
days or 37 months. Data relates to actual (market) prices of ten Eurobonds from all BRIC
countries (Brazil, Russia, India and China). BRIC member countries together encompass over
25% of the world's land coverage, 40% of the world's population, and about 25% of the global
GDP in 2010, with significant increases in global GDP share expected over the next four
decades.
The sovereign bond ratings from Moody's for the BRIC countries are: Brazil (BBB −), Russia
(BBB +), India (BBB −) and China (A +)8. There are similarities as well as some differences
between the stock exchanges of the BRIC countries. According to Table 1, China is ranked
first and Russia last among BRIC countries in terms of market capitalization, market
capitalization to GDP, the MSCI Emerging markets index weights and the S&P/IFC EM Index
weights. Brazil and India are ranked in between. China is also ranked first in terms of GDP
growth, with India second, Russia third and Brazil last. Table 1.
BRIC countries' stock exchanges.
Country
GDP growth (%) Exchange
Market capitalization
Market capitalization to GDP (%)
MSCI emerging markets index weights
S&P/IFC EM index weights
Brazil 5.08 BM & FBOVESPA
1,337,248 74.26 16.90% 11.99%
Russia 5.60 MICEX 736,307 69.99 6.30% 6.45%
India 6.07 Bombay SE 1,306,520 90.01 7.50% 7.39%
China 9.00 Shanghai SE 2,704,778 100.46 17.90% 17.29%
Notes. Table 1 reports the name of the major stock exchange of each of the BRIC countries, as well as the market
capitalization, market capitalization to GDP, the country-weights in the MSCI Emerging markets index, and the
S&P/IFC EM Index weights. Market capitalization is in $ millions. Table 1 depicts data for the year 2010 coming from
the WFSE (World Federation of Stock Exchanges) historical statistics; only the GDP growth (as a %) is provided by
the World Bank.
Table 2 provides the symbol, description, country of origin, expiry year as well as an
indication for either actual (market) or theoretical prices. Each country's Eurobonds market is
analyzed by three Eurobonds with the only exception being India for which only one Eurobond
is employed. The expiry year differs across these Eurobonds. Two Eurobonds expired in late
2010, one in 2011, one in 2012, one in 2013, three expired in 2014, one will expire in 2015
and one in 2016. Daily bond prices have been used to estimate Eurobonds' monthly risk and
monthly jumps, as the monthly frequency is appropriate not for traders but for long-term
investors (mostly, pension funds) in Eurobond markets.
Table 2.
BRIC countries' Eurobonds.
Symbol Description Country Expiry year The/Act
B14act brazil_7_14_2014 Brazil 2014 Act.
B14the brazil_7_14_2014 Brazil 2014 The.
B11act brazil_8_7_2011 Brazil 2011 Act.
B11the brazil_8_7_2011 Brazil 2011 The.
B12act brazil_1_11_2012 Brazil 2012 Act.
B12the brazil_1_11_2012 Brazil 2012 The.
R13act russian_agri_5_16_2013 Russia 2013 Act.
R13thw russian_agri_5_16_2013 Russia 2013 The.
R10act,1 bank_of_moscow_11_26_2010 Russia 2010 Act.
R10the,1 bank_of_moscow_11_26_2010 Russia 2010 The.
R10act,2 bank_of_moscow_11_29_2010 Russia 2010 Act.
R10the,2 bank_of_moscow_11_29_2010 Russia 2010 The.
I16act ntpc_india_3_2_2016 India 2016 Act.
I16the ntpc_india_3_2_2016 India 2016 The.
C14act,1 china_dev_bank_10_8_2014 China 2014 Act.
C14the,1 china_dev_bank_10_8_2014 China 2014 The.
C14act,2 exim_china_7_29_2014 China 2014 Act.
C14the,2 exim_china_7_29_2014 China 2014 The.
C15act china_dev_bank_10_15_2015 China 2015 Act.
C15the china_dev_bank_10_15_2015 China 2015 The.
Notes. Table 2 reports the symbol, description, country, expiry year, and the indication of actual or theoretical prices
series. Theoretical prices are retrieved as in Section 3.1.
2.2. Descriptive analysis
Return is the logarithmic difference between two consecutive prices. Table 3 presents
descriptive statistics (mean, standard deviation, skewness and kurtosis) as well as the
normality hypothesis results (CVM-test and QQ-test) for returns. The mean return as well as
the standard deviation are the highest for the Russian Eurobond, compared to others.
Skewness and kurtosis values indicate the distributions of returns in most of the BRIC
Eurobonds are skewed to the right (skewness higher than zero) and leptokurtic (kurtosis
higher than three). However, the CVM and LB normality tests do not reject the null hypothesis
of normality for most of the BRIC Eurobonds.
Table 3.
Returns–descriptive statistics.
Mean St. dev. Skew. Kurt. CVM QQ
Mean St. dev. Skew. Kurt. CVM QQ
B14act − 2.91e − 4 0.0238 0.1424 6.05 0.1523 21.43
B14the − 3.24e − 4 0.0203 − 0.4529 3.33 0.1341 36.13
B11act − 6.26e − 4 0.0131 − 0.1145 3.22 0.1052 19.35
B11the − 7.32e − 4 0.0204 − 0.852 3.58 0.1141 24.94
B12act − 0.0015 0.0184 − 0.2023 5.93 0.2604 21.47
B12the − 0.0020 0.0183 − 1.02 3.18 0.300 44.28⁎
R13act 2.65e − 4 0.0642 − 1.90 9.37 0.5757⁎ 10.99
R13the 6.94e − 4 0.0171 − 0.4896 3.20 0.0813 13.47
R10act,1 − 2.94e − 4 0.0576 − 1.34 8.72 0.7759⁎ 9.65
R10the,1 − 0.0019 0.0208 − 1.04 3.87 0.4109 78.83⁎
R10act,2 1.16e − 4 0.0302 − 1.29 7.84 0.5553⁎ 16.41
R10the,2 − 4.11e − 4 0.0167 − 0.7762 4.05 0.2852 13.62
I16act 0.0014 0.0262 − 0.5784 4.82 0.0688 14.25
I16the 0.0018 0.0234 0.9926 5.03 0.0695 12.61
C14act,1 0.0018 0.0255 − 1.56 9.04 0.2282 22.71
C14the,1 0.0023 0.0211 − 0.1093 3.71 0.0541 21.56
C14act,2 0.0022 0.0461 0.2685 12.19 0.5929⁎ 18.03
C14the,2 0.0020 0.0189 − 0.0681 3.58 0.0933 23.19
C15act 0.0029 0.1711 0.0935 15.14 1.56⁎ 14.86
C15thw 0.0022 0.0256 0.1171 4.07 0.0585 29.32
Notes. The mean, standard deviation, skewness and kurtosis values as well as the CVM and QQ test statistics are
reported. All descriptive statistics are reported for either actual returns (indicated by act) or theoretical returns
(indicated by the).
⁎
Indicates significance in 5% significance level.
3. Empirical methodology
Bond prices employed, used both actual market prices and theoretical prices. Returns are
produced for both actual and theoretical prices. Monthly point estimates of volatility are
estimated through three groups of estimators: realized volatility, range and realized range-
based volatility. Then, monthly jumps are detected from two different detection schemes.
3.1. Bond pricing
Using bond prices is more reliable than using yields. This is because yields are retrieved from
actual bond prices and may be depended on different maturities and coupons. The pricing
model involves fiting a smooth discount function to information obtained from observed prices
of straight bonds with various coupons and maturities by estimating the coefficients for a
linear combination of smooth approximating functions forming a cubic spline. Any coupon
bond price maturing at par value and paying a coupon at timei can be expressed as:
equation(1)
where P = clean price or the price quoted in the market (as % of par
value), C = coupon, Ri = discount rate applicable for period i with T as the final maturity date.
Replacing by, returns
equation(2)
The discount function di can be expressed as a combination of smooth approximating
functions and defines the present value of 1 unit of any numeraine receivable
in i years. McCulloch, 1971 and McCulloch, 1975 suggested that the discount function di can
be expressed as:
equation(3)
where kfi(i) functions are chosen (the value of k varying with the exact model) to estimate d(i) by a cubic spline and the aj are the estimated parameters of the linear regression. The fi(i), (j = 1,..., k) are chosen so that fj(0) = 0 to force d(0) = 1 and to enable it to be smooth and
monotonically nonincreasing. Substituting di with d(i) in the P + AI equation, the price of a
bond maturing in T months and paying a coupon at time i can be expressed as follows:
equation(4)
In case of a discrete time, it is employed a discount function with two cubic splines, k = 5 and
∑f = 1kaifi(i) = ai + βi2 + γi3 + γ1DV1i(i − t1⁎)3 + γ2DV2i(i − t2⁎)3. Then the discount factor is
equation(5)
where DV1 and DV2 are dummy variables shifting the cubic term of the polynomial for time
points. These are the knot points for the cubic spline. When D(i) is substituted in
the P + AI equation and an error term is added then, the final form of the pricing model is:
equation(6)
where P is the clean price, AI is the accrued coupon, T is the total number of coupons
left, h is the date to the first coupon, i = 1 is the number of coupons left to maturity (up to T)
and hi is the date of the last cash flow. DV represents dummy variables representing the
spline knots if time left to maturity of the bond is greater than t(⋅) *. Taking a large cross section
of bonds in a market at a point in time with differing market prices, of diverse coupons and
times to maturities and using regression allows the estimation of a, β, γ, γ1, and γ2 using the
last equation. The error term in the regression ensures that random effects are captured.
Repeating this exercise over time ensures a time series of a, β, γ, γ1, andγ2.
The estimates of bond prices via the above bond pricing method return the so-called
theoretical bond prices, which are indicated as ‘the’, and market prices are indicated as ‘act’. The risk and jumps of BRIC Eurobonds are compared across ten Eurobonds and the four
countries as well as across twelve volatility estimators which are split in three groups (realized
volatility, range, and realized range-based volatility). Each group consists of four estimators.
3.2. Realized volatility estimators
All realized volatility estimators provide monthly point estimates by using daily
returns. Andersen et al. (2001) suggested the unrestricted realized volatility estimator (RVt(m)):
equation(7)
where t is the indication of the month, i indicates the trading day in a specific t month and m is
the number of trading days per month across all realized volatility and range estimators. This
notation is consistent across all volatility estimators. Barndorff-Nielsen, Hansen, Lunde, and
Shephard (2011)theoretically and empirically examined the realized bipower variation
(BPVt(m)). In literature, this estimator is employed to detect jumps because the realized
bipower variation has no jumps.
equation(8)
where μp = E(|Z|p) is the mean of the pth absolute moment of a standard normal
distribution. Hansen, Large, and Lunde (2008) constructed a moving average-based volatility
estimator that uses the first order MA(1) residuals (RVt(ma. adj1)):
equation(9)
where , and is estimated for each the month. The aim is to reduce the
autocorrelation in daily returns. This was evident in the QQ-test statistic values in Table 3 as
explained in Section 2.2. Hansen et al. (2008) with Bandi, Russell, and Yang (2008) also
proposed a moving average-based volatility estimator that uses the q order MA(q) residuals.
The q order is selected according to the AIC criterion. This estimator should be more accurate
in case of more than one MA orders (RVt(ma. adj2)):
equation(10)
where and RVt(m) is the unrestricted realized volatility estimator.
3.3. Range-based estimators
Range is the difference between the highest and lowest price. Range estimators are split into
two categories: monthly ranges, and realized range-based volatility estimators. The monthly
range estimators use the highest and lowest monthly prices per month and symbolized
as MRt. The range estimators, also estimated monthly, using the highest and lowest daily
prices per day are entitled as realized range-based volatility estimators and symbolized
as RRt. The present paper examines four monthly ranges as well as their corresponding four
realized range-based estimators. These estimators are: Parkinson, Garman & Klasss, Rogers
& Satchell, and Yang & Zhang; either monthly or realized.
3.3.1. Monthly ranges
Parkinson (1980) defined and empirically analyzed the range estimator. That is why the first
version of a range estimator is entitled as Parkinson range. As far as the sampling frequency
of the estimator is monthly, it can be called monthly Parkinson estimator:
equation(11)
where Ph,t(m) is the highest monthly price (the highest price of the month) and Pl,t(m) is the
lowest monthly price (the lowest price of the month). Garman and Klass (1980) extended the
Parkinson estimator to:
equation(12)
where n is the total number of monthly observations, Pc,t(m) is the monthly close price (the
closing price per month) and Po,t(m) is the monthly open price (the opening price per
month). Rogers and Satchell (1991) extended the Parkinson estimator, in a similar way
to Garman and Klass (1980) estimator, via incorporating monthly open and close prices apart
from the monthly high and low prices:
equation(13)
Yang and Zhang (2000) incorporated a term for the closed market variance (that is the over-
month variance; i.e. a month-effect). So, the monthly Yang and Zhang estimator is defined as:
equation(14)
where n is the number of months,
, and . MRt(RS) is the monthly Rogers &
Satchell range estimator, Pc,t(T) is the monthly close price, Po,t(T) is the monthly open
price, Ph,t(T) is the monthly high price, Pl,t(T ) is the monthly low price, is the average
monthly close price (average value of all monthly close prices) and Pō,t(T) is the average
monthly open price (average value of all monthly open prices).
3.3.2. Realized range-based
When the four range-based estimators are estimated monthly via daily data, they are known
as realized range-based estimators. The realized Parkinson range-based volatility estimator is
suggested in Martens and van Dijk (2007) (RRt(Par)) as:
equation(15)
where m is the number of trading days per month, hi,m = ln(Ph(i, m)), and li ,m = ln(Pl(i, m)) are
the within the i-th daily interval (per day; daily) high and low logarithmic prices. The realized
Garman and Klass range-based estimator (RRt(GK)) is:
equation(16)
where n is the number of
months, Ri,m,1 = [ln(Ph(i, m)/Pl(i, m))]2, Ri,m,2 = ln(Pc(i, m)/Po(i, m)),Ri,m,3 = ln(Ph(i, m) ⋅ Pl(i, m)/Po2(
i, m)), Ri,m,4 = ln(Ph(i, m)/Po(i, m)), and Ri,m,5 = ln(Pl(i, m)/Po(i, m)).Rogers and Satchell (1991)'s
estimator can also be estimated as a realized range-based volatility estimator. So, the
realized Rogers and Satchell range-based estimator (RRt(RS)) is given by:
equation(17)
where = ln(Ph(i, m)/Pc(i, m )), = ln(Ph(i, m)/Po(i, m)), = ln(Pl(i, m)/Pc(i, m )),
and = ln(Pl(i, m)/Po(i, m)). Finally, the realized Yang and Zhang range-based volatility
estimator (RRt(YZ)) is given by the following equation:
equation(18)
where Ri,m,1⁎ = ln(Po(i, m)/Pc(i, m) − Pō(i, m )),
, , and .
3.4. Jumps
The detection scheme employed to detect jumps on the monthly volatility series was
introduced in Ait-Sahalia and Jacod (2009) and further examined in Ait-Sahalia and Jacod
(2011) and Ait-Sahalia, Jacod, and Li (2012).
All three papers detect a jump in volatility series when there is a significant difference
between the realized quarticity of a specific sampling frequency and a multiple of it. The
critical value for the test of this jump detection scheme is
equation(19) Fa=2−Φa⋅(V)1/2I( |V t|>c 1 )V t+I( |V t|<c 2 )Vt
where c1 = 0.95, c2 = 0.05, and Vt is any of either realized volatility estimators or range
estimators explained in the previous subsection,
ri ,m,1 = ln(Pc,1(i, m)/Pc,1(i − 1, m)), ri ,m,2 = ln(Pc,2(i, m)/Pc,2(i − 1, m)), Pc,1(i, m) is the daily close
prices, and Pc,2(i, m) is the daily close prices for the multiple of the first sampling (i.e. daily)
frequency (inPc,1(i, m)). The standardized test statistic is
where . There are jumps for a month, when S < Fa. The empirical results
reported below are relied on a significance level of a = 5%.
equation(20)
JMt=m ax( |V t−(V)1/2|,0)
The jump part (JMt) of any Vt estimator is estimated as in Andersen, Bollerslev, Frederiksen,
and Nielsen (2010) 9. Jump frequency (JFt) is the frequency of occurrence of monthly jump
upon the total number of months in sample; so, it is the number of months that jumps are
detected, is expressed as a percentage to the total number of months for the examined
(either before or after) time period. The indicator of the existence of at least one jump per
month can be depicted as: Bt = I(JMt ≠ 0).
4. Empirical findings
All measures are based on average values of monthly point estimates of risk and jumps. Risk
is measured via the mean magnitude of risk ( ) and the mean Sharpe ratio ( ) as well.
Results for risk are reported inTable 6 and Table 7 accordingly. The significance of jumps is
measured via the mean magnitude of jumps ( ), the mean ratio of the magnitude of the
jump component of risk relative to the magnitude of the continuous component ( ), and the
average frequency of jump occurrence ( ). Results for jumps are reported
in Table 8, Table 9 and Table 10 respectively. Table 4.
Risk–descriptive statistics–skewness and kurtosis.
RVt(
m) BPVt(m)
RVt(ma.
adj1) RVt(ma.
adj2) MRt(
Par) MRt(GK)
MRt(RS)
MRt(YZ)
RRt(
Par) RRt(
GK) RRt(RS)
RRt(
YZ)
B14ac
t 5.54 (32.44)
5.59 (32.85)
5.60 (32.86)
5.56 (32.56)
5.38 (31.07)
− 2.12 (8.41)
− 2.11 (7.63)
− 2.06 (7.36)
5.57 (32.7)
5.57 (32.69)
5.60 (32.91)
5.54 (32.47)
B14th
e 2.75 (9.69)
3.18 (12.08)
3.05 (11.34)
4.12 (20.42)
2.49 (8.16)
0.2087 (1.99)
0.9710 (3.54)
0.9853 (3.57)
2.74 (9.59)
2.75 (9.61)
3.62 (16.37)
3.02 (11.46)
B11ac
t 29.46 (30.21)
5.53 (30.62)
6.51 (30.69)
6.43 (30.67)
5.16 (29.46)
− 1.14 (5.53)
− 1.62 (6.51)
− 1.60 (6.43)
5.27 (30.43)
1.06 (5.36)
3.30 (20.48)
3.40 (20.56)
B11th
e 1.36 (4.04)
1.86 (6.53)
2.04 (7.58)
2.29 (8.48)
1.11 (2.75)
0.3008 (2.49)
0.472 (3.02)
0.4756 (3.03)
2.57 (9.88)
1.25 (4.00)
0.3362 (5.75)
1.34 (6.86)
B12ac
t 5.28 (30.4
5.37 (31.1
5.36 (31.11)
5.46 (31.79)
5.25 (30.0
− 1.45 (6.50)
− 1.38 (5.09)
− 1.35 (4.98)
5.38 (31.
5.37 (31.
5.53 (32.4
5.30 (30.
RVt(
m) BPVt(m)
RVt(ma.
adj1) RVt(ma.
adj2) MRt(
Par) MRt(GK)
MRt(RS)
MRt(YZ)
RRt(
Par) RRt(
GK) RRt(RS)
RRt(
YZ)
6) 2) 9) 20) 17) 2) 61)
B12th
e 5.12 (29.12)
5.51 (32.23)
5.53 (32.40)
5.67 (33.46)
4.16 (21.48)
− 0.2363 (2.12)
− 0.5065 (2.84)
− 0.4864 (2.78)
5.12 (29.13)
5.12 (29.09)
5.56 (32.58)
5.60 (32.9)
R13ac
t 2.92 (10.06)
2.92 (9.96)
2.93 (10.03)
2.96 (10.15)
4.42 (23.22)
− 1.72 (4.86)
− 1.54 (4.09)
− 1.55 (4.15)
2.93 (10.10)
2.93 (10.11)
2.85 (9.42)
2.91 (10.02)
R13th
e 4.60 (24.85)
4.34 (22.37)
4.24 (21.46)
3.06 (14.17)
5.24 (30.15)
− 0.379 (3.27)
0.6699 (5.33)
0.7043 (5.44)
4.41 (23.24)
4.43 (23.37)
4.15 (20.95)
4.60 (24.84)
R10a
ct,1 5.23 (29.75)
5.38 (30.94)
5.38 (30.96)
5.56 (32.54)
4.21 (21.20)
− 2.73 (10.15)
− 2.52 (8.84)
− 2.55 (9.00)
5.20 (29.60)
5.18 (29.42)
5.27 (30.31)
5.23 (29.78)
R10the,1
3.42 (15.21)
2.45 (8.00)
2.58 (8.84)
3.13 (13.02)
4.67 (25.62)
0.0042 (2.15)
− 0.9499 (6.57)
− 0.9041 (6.41)
2.25 (8.18)
1.43 (4.84)
− 0.4806 (2.92)
1.72 (6.52)
R10act,2
3.87 (18.12)
5.30 (30.46)
5.39 (31.20)
4.98 (27.79)
3.29 (13.62)
− 2.42 (8.47)
− 2.07 (6.41)
− 2.08 (6.43)
3.74 (17.09)
3.74 (17.08)
3.87 (18.13)
3.87 (18.11)
R10the,2
2.52 (9.55)
2.76 (9.53)
2.91 (11.36)
3.96 (18.82)
1.69 (4.46)
0.4995 (2.76)
0.5122 (2.56)
0.51 (2.56)
2.70 (10.71)
2.70 (10.71)
3.03 (13.59)
2.52 (9.67)
I16act 2.63 (9.49)
2.81 (10.46)
2.80 (10.27)
2.92 (12.09)
1.75 (5.04)
− 0.6449 (2.80)
− 0.6546 (2.59)
− 0.65 (2.59)
2.62 (9.53)
2.63 (9.61)
2.74 (10.37)
2.69 (9.80)
I16the 5.53 (32.40)
5.55 (32.51)
5.50 (32.15)
1.11 (3.33)
5.33 (30.69)
− 0.366 (2.06)
1.90 (9.91)
1.97 (10.25)
5.49 (32.13)
5.50 (32.16)
5.37 (31.19)
5.54 (32.44)
C14act,1
3.57 (16.04)
0.3279 (2.67)
0.2817 (2.83)
0.2869 (2.83)
3.89 (17.64)
4.03 (19.38)
4.03 (19.38)
3.87 (20.21)
3.86 (17.48)
3.87 (17.58)
3.81 (17.25)
3.90 (17.70)
C14t
he,1 2.59 (5.72)
3.11 (2.25)
2.97 (2.31)
2.63 (2.32)
1.92 (9.95)
− 0.4575 (12.69)
− 0.3232 (11.83)
− 0.319 (10.57)
2.71 (10.56)
2.71 (10.59)
3.70 (18.33)
2.94 (12.36)
C14act,2
5.42 (32.34)
1.83 (30.8)
− 0.2679 (30.76)
− 0.2633 (13.05)
5.53 (31.47)
5.34 (9.25)
5.34 (2.68)
3.12 (2.63)
5.49 (32.01)
5.49 (32.03)
5.31 (30.53)
5.50 (32.14)
C14the,2
3.07 (12.34)
3.31 (13.35)
3.28 (13.3)
3.43 (15.28)
2.50 (8.65)
− 0.4076 (2.22)
− 0.0195 (2.09)
− 0.0143 (2.10)
3.16 (12.84)
3.17 (12.87)
4.20 (21.48)
3.44 (15.12)
C15ac
t 4.41 (21.5
5.21 (29.3
5.25 (29.78)
5.75 (34.03)
3.85 (15.9
1.72 (12.9)
4.23 (24.2
4.32 (24.5
5.74 (33.
5.74 (33.
3.98 (17.8
5.74 (33.
RVt(
m) BPVt(m)
RVt(ma.
adj1) RVt(ma.
adj2) MRt(
Par) MRt(GK)
MRt(RS)
MRt(YZ)
RRt(
Par) RRt(
GK) RRt(RS)
RRt(
YZ)
9) 7) 8) 4) 4) 99) 99) 6) 98)
C15th
e 5.75 (34.03)
5.75 (34.03)
5.75 (34.03)
5.75 (34.03)
5.74 (34.00)
− 2.72 (13.31)
4.99 (28.58)
5.08 (29.21)
5.75 (34.03)
5.75 (34.03)
2.52 (8.20)
5.75 (34.03)
Notes. Table 4 reports the descriptive statistics of skewness (outside brackets) and kurtosis (within brackets) for the
risk estimates of both actual and theoretical BRIC Eurobond prices. Risk estimates are split into three groups:
realized volatility, monthly range-based, and realized range estimates.
Table 5.
Risk–descriptive statistics–CVM and QQ tests.
RVt(m)
BPVt(
m) RVt(ma.
adj1) RVt(ma.
adj2) MRt(P
ar) MRt(
GK) MRt(
RS) MRt(Y
Z) RRt(P
ar) RRt(G
K) RRt(
RS) RRt(Y
Z)
B14a
ct 2.02⁎
(3.29)
2.14⁎
(2.44)
2.14⁎
(2.43) 2.17⁎
(2.77) 2.12⁎
(3.79) 0.3587 (52.62⁎)
0.4924⁎
(44.58⁎)
0.4829⁎
(45.26⁎)
2.07⁎
(2.94) 2.07⁎
(2.95) 2.16⁎
(2.65)
2.03⁎
(3.24)
B14th
e 1.20⁎
(18.23)
1.47⁎
(18.54)
1.41⁎
(18.29) 1.37⁎
(7.97) 1.13⁎
(28.92)
0.0634 (28.79)
0.1210 (16.68)
0.1243 (16.77)
1.21⁎
(18.24)
1.21⁎
(18.26)
1.37⁎
(13.95)
1.28⁎
(17.42)
B11a
ct 1.48⁎
(5.33)
1.59⁎
(4.81)
1.60⁎
(4.71) 1.72⁎
(4.65) 1.53⁎
(6.08) 0.0983 (47.69⁎)
0.1757 (39.36⁎)
0.1730 (39.52⁎)
1.47⁎
(5.05) 0.3823 (27.28)
1.40⁎
(12.78)
1.21⁎
(13.09)
B11th
e 0.5398⁎
(27.12)
0.5460⁎
(18.94)
0.5384⁎(16.7)
0.6118⁎
(19.63)
0.5574⁎
(54.74⁎)
0.0417 (24.95)
0.0390 (51.52⁎)
0.0390 (51.35⁎)
1.18⁎
(19.06)
0.4856⁎
(15.73)
0.8090⁎
(30.14)
0.8415⁎
(31.15)
B12a
ct 1.63⁎
(6.04)
1.70⁎
(4.51)
1.69⁎
(4.53) 2.04⁎
(4.18) 1.82⁎
(5.63) 0.1307 (40.01⁎)
0.1941 (53.28⁎)
0.1912 (53.66⁎)
1.72⁎
(5.10) 1.72⁎
(5.14) 1.94⁎
(3.46)
1.65⁎
(5.91)
B12th
e 1.67⁎
(4.55)
2.02⁎
(2.18)
2.04⁎
(2.10) 2.31⁎
(1.20) 1.28⁎
(8.59) 0.0658 (93.36⁎)
0.0392 (27.25)
0.0390 (27.68)
1.65⁎
(4.57) 1.65⁎
(4.61) 2.09⁎
(2.01)
2.19⁎
(1.70)
R13a
ct 1.76⁎
(36.07)
1.92⁎
(32.6)
1.92⁎
(32.29) 1.84⁎
(33.77) 1.72⁎
(18.15)
0.8213⁎
(124.67⁎)
0.7648⁎
(129.35⁎)
0.7619⁎
(128.22⁎)
1.79⁎
(35.86)
1.79⁎
(35.85)
1.79⁎
(35.60)
1.76⁎
(36.16)
R13th
e 1.31⁎
(8.92)
1.38⁎
(11.27)
1.38⁎
(11.77) 0.5907⁎
(15.56)
1.46⁎
(4.25) 0.0435 (34.10)
0.0939 (15.95)
0.0963 (15.79)
1.26⁎
(10.31)
1.27⁎
(10.15)
1.21⁎
(12.38)
1.31⁎
(8.84)
R10a
ct,1 1.99⁎
(6.981.10⁎
(4.251.00⁎
(4.25) 0.9950⁎(3.67)
2.14⁎
(17.42.36⁎
2.37⁎
2.31⁎
(54.72.11⁎
(7.46) 1.97⁎
(7.08) 1.85⁎
2.09⁎
(7.04)
RVt(m)
BPVt(
m) RVt(ma.
adj1) RVt(ma.
adj2) MRt(Par)
MRt(
GK) MRt(
RS) MRt(YZ)
RRt(Par)
RRt(GK)
RRt(
RS) RRt(YZ)
) ) 1) (54.07⁎)
(55.83⁎)
1⁎) (5.27)
R10the,1
1.25⁎
(14.18)
1.28⁎
(22.74)
1.26⁎
(21.40) 1.31⁎
(16.33) 1.40⁎
(5.33) 0.0417 (131.96⁎)
0.0661 (88.88⁎)
0.0636 (89.66⁎)
0.7855⁎
(20.93)
0.3801 (23.56)
0.3074 (30.09)
0.6674⁎
(24.77)
R10a
ct,2 1.99⁎
(17.59)
2.08⁎
(4.64)
2.15⁎
(3.80) 2.13⁎
(6.52) 1.98⁎
(26.22)
0.8076⁎
(74.97⁎)
0.7547⁎
(84.32⁎)
0.7508⁎
(84.14⁎)
1.98⁎
(19.59)
1.98⁎
(19.60)
1.99⁎
(17.55)
1.99⁎
(17.60)
R10the,2
0.8784⁎
(10.95)
1.29⁎
(13.44)
1.13⁎
(13.80) 1.46⁎
(12.82) 0.9379⁎
(15.09)
0.0532 (34.18)
0.0685 (52.48⁎)
0.0687 (52.30⁎)
0.8867⁎
(11.90)
0.8864⁎
(11.96)
0.8051⁎
(11.28)
0.8326⁎
(11.17)
I16act 0.8435⁎
(20.13)
1.01⁎
(14.40)
1.04⁎
(14.60) 0.7566⁎
(17.01)
0.6243⁎
(27.90)
0.1806 (169.59⁎)
0.1997 (134.23⁎)
0.1986 (134.28⁎)
0.8189⁎
(20.42)
0.8229⁎
(20.30)
0.9382⁎
(18.45)
0.8857⁎
(19.67)
I16the 1.87⁎
(1.61)
1.94⁎
(1.68)
1.85⁎
(2.04) 0.3783 (48.79⁎)
1.84⁎
(2.30) 0.0647 (34.10)
0.1365 (14.06)
0.1429 (13.81)
1.80⁎
(1.79) 1.81⁎
(1.77) 1.66⁎
(2.60)
1.89⁎
(1.55)
C14a
ct,1 1.49⁎
(11.74)
1.26⁎
(8.96)
1.25⁎
(8.88) 0.5929⁎
(15.15)
1.26⁎
(14.78)
0.0573 (64.58⁎)
0.0251 (63.17⁎)
0.0258 (63.73⁎)
1.42⁎
(12.17)
1.43⁎
(12.09)
1.31⁎
(11.42)
1.48⁎
(11.63)
C14the,1
0.6690⁎
(17.92)
0.9414⁎
(12.14)
0.9114⁎
(12.65)
0.7668⁎
(17.31)
0.6970⁎
(28.39)
0.0754⁎
(53.16⁎)
0.1273 (38.22⁎)
0.1262 (38.22⁎)
1.00⁎
(12.15)
1.00⁎
(12.06)
1.25⁎
(7.17)
1.05⁎
(10.69)
C14a
ct,2 2.03⁎
(1.82)
0.1119 (3.21)
0.0352 (3.24)
0.0382 (18.13)
2.11⁎
(2.43) 2.03⁎
(41.10⁎)
2.02⁎
(45.99⁎)
0.9921⁎
(47.23⁎)
2.07⁎
(2.07) 2.07⁎
(2.05) 2.01⁎
(3.50)
2.09⁎
(2.00)
C14t
he,2 0.9507⁎
(12.56)
1.20⁎
(10.32)
1.16⁎
(10.22) 1.06⁎
(11.22) 0.8338⁎
(28.30)
0.1027⁎
(52.59⁎)
0.0378 (29.94)
0.0371 (29.93)
0.7091⁎
(17.02)
0.7126⁎
(16.88)
0.8882⁎
(9.12)
0.741⁎
(14.35)
C15a
ct 2.58⁎
(10.39)
2.58⁎
(10.37)
2.58⁎
(7.01) 2.82⁎
(7.53) 2.41⁎
(7.24) 0.5993⁎
(2.94)
0.9753⁎
(2.71)
0.9788⁎
(0.4744)
2.76⁎
(0.4648)
2.76⁎
(0.4648)
1.96⁎
(17.28)
2.76⁎
(0.4677)
C15th
e 2.81⁎
(0.6857)
2.82⁎
(0.6880)
2.82⁎
(0.6875)
2.81⁎
(0.6868)
2.74⁎
(0.6826)
0.2416 (16.5
1.12⁎
(5.04
1.20⁎
(4.58) 2.85⁎
(0.6829)
2.85⁎
(0.6829)
1.59⁎
(20.
2.85⁎
(0.682)
RVt(m)
BPVt(
m) RVt(ma.
adj1) RVt(ma.
adj2) MRt(Par)
MRt(
GK) MRt(
RS) MRt(YZ)
RRt(Par)
RRt(GK)
RRt(
RS) RRt(YZ)
0) ) 21)
Notes. Table 5 reports the descriptive statistics of CVM (outside brackets) and QQ (within brackets) normality tests
for the risk (volatility) estimates of both actual and theoretical BRIC Eurobond prices. Risk (volatility) estimates are
split into three groups: realized volatility, monthly range, and realized range estimates.
⁎
Indicates significance in a 5% significance level.
Table 6.
Average magnitude of risk ( ).
RVt(m)
BPVt(
m) RVt(ma. adj1)
RVt(ma. adj2)
MRt(P
ar) MRt(G
K) MRt(R
S) MRt(Y
Z) RRt(P
ar) RRt(G
K) RRt(R
S) RRt(Y
Z)
B14a
ct 8.54e − 4
9.32e − 4
9.31e − 4
6.12e − 4
6.50e − 4
5.39e − 5
3.90e − 5
3.60e − 5
5.47e − 4
5.58e − 4
4.94e − 4
8.15e − 4
B14t
he 0.0013
0.0020
0.0019
7.40e − 4
5.49e − 4
5.54e − 5
4.02e − 5
3.71e − 5
8.18e − 4
8.42e − 4
7.36e − 4
0.0013
B11a
ct 3.06e − 4
3.88e − 4
3.85e − 4
2.16e − 4
1.60e − 4
5.20e − 5
3.77e − 5
3.47e − 5
2.06e − 4
3.30e − 4
1.27e − 4
2.39e − 4
B11t
he 7.04e − 4
9.10e − 4
9.17e − 4
4.42e − 4
3.29e − 4
5.44e − 5
3.95e − 5
3.64e − 4
3.17e − 4
3.50e − 4
1.22e − 4
3.34e − 4
B12a
ct 4.20e − 4
4.34e − 4
4.34e − 4
3.24e − 4
2.76e − 4
5.29e − 5
3.83e − 5
3.53e − 4
3.21e − 4
3.26e − 4
2.91e − 4
4.80e − 4
B12t
he 0.0013
0.0020
0.0020
7.91e − 4
3.85e − 4
5.36e − 5
3.88e − 5
3.58e − 5
8.88e − 4
9.13e − 4
0.0010
0.0022
R13act
0.0015
0.0015
0.0016
0.0013
0.0021
4.79e − 5
3.49e − 5
3.22e − 5
0.0011
0.0012
8.62e − 4
0.0016
R13t
he 8.99e − 4
0.0012
0.0012
4.69e − 4
3.65e − 4
5.18e − 5
3.77e − 5
3.47e − 5
6.27e − 4
6.38e − 4
5.29e − 4
9.75e − 4
R10act,1
0.0018
0.0025
0.0025
0.0013
0.0015
4.87e − 5
3.54e − 5
3.26e − 5
0.0012
0.0025
0.0019
0.0038
R10t
he,1 0.0014
0.0023
0.0022
7.25e − 4
4.59e − 4
5.04e − 5
3.65e − 5
3.37e − 5
3.01e − 4
0.0035
0.0029
0.0052
R10act,2
0.0014
0.0019
0.0019
9.49e − 4
0.0011
4.97e − 5
3.61e − 5
3.33e − 5
6.94e − 4
7.17e − 4
5.15e − 4
0.0011
R10the,2
4.46e − 4
8.66e − 4
7.35e − 4
2.88e − 4
1.35e − 4
5.03e − 5
3.65e − 5
3.36e − 5
3.41e − 4
3.51e − 4
2.85e − 4
5.06e − 4
I16act 8.02e − 4
0.0011
0.0011
5.78e − 4
3.80e − 4
4.78e − 5
3.46e − 5
3.19e − 5
4.51e − 4
4.68e − 4
4.45e − 4
7.89e − 4
I16th
e 0.0020
0.0031
0.0030
9.00e − 4
8.05e − 4
5.09e − 5
3.69e − 5
3.40e − 5
0.0012
0.0013
0.0011
0.0020
C14act,1
8.50e − 4
9.24e − 4
9.24e − 4
4.67e − 4
4.07e − 4
4.91e − 5
3.55e − 5
3.28e − 5
6.34e − 4
6.42e − 4
5.09e − 4
9.67e − 4
C14t
he,1 8.29e − 4
0.0012
0.0012
7.24e − 4
3.90e − 4
4.96e − 5
3.60e − 5
3.32e − 5
5.23e − 4
5.29e − 4
4.79e − 4
8.31e − 4
RVt(m)
BPVt(
m) RVt(ma. adj1)
RVt(ma. adj2)
MRt(Par)
MRt(GK)
MRt(RS)
MRt(YZ)
RRt(Par)
RRt(GK)
RRt(RS)
RRt(YZ)
C14act,2
9.81e − 4
0.0011
0.0012
6.94e − 4
5.85e − 4
4.89e − 5
3.54e − 5
3.27e − 5
9.51e − 4
9.74e − 4
7.93e − 4
0.0015
C14act,2
7.50e − 4
0.0012
0.0011
6.23e − 4
3.51e − 4
5.02e − 5
3.64e − 5
3.35e − 5
6.41e − 4
6.50e − 4
5.71e − 4
9.76e − 4
C15act
0.0014
0.0018
0.0018
4.21e − 4
6.62e − 4
4.96e − 5
3.58e − 5
3.30e − 5
0.0192
0.0198
6.52e − 4
0.0277
C15t
he 0.0258
0.0468
0.0511
0.0208
0.0051
4.96e − 5
3.64e − 5
3.36e − 5
0.0185
0.0191
4.64e − 4
0.0268
Notes. Table 6 reports the average magnitude of risk (mean volatility) ( ) for both actual and theoretical BRIC
Eurobond prices. Risk (volatility) estimates are split into three groups: realized volatility, monthly range, and realized
range estimates. All average values reported, are t-test significant in a 5% significance level.
Table 7.
Average Sharpe ratio ( ).
RVt(m)
BPVt(
m) RVt(ma.
adj1) RVt(ma.
adj2) MRt(Par)
MRt(
GK) MRt(
RS) MRt(
YZ) RRt(Par)
RRt(GK)
RRt(RS)
RRt(YZ)
B14a
ct − 0.3401
− 0.3117
− 0.3119
− 0.4747
− 0.4467
− 5.39
− 7.44
− 8.08
− 0.5314
− 0.5206
− 0.5879
− 0.3566
B14th
e − 0.2557
− 0.1600
− 0.1679
− 0.4378
− 0.5902
− 5.85
− 8.05
− 8.74
− 0.3962
− 0.3850
− 0.4405
− 0.2446
B11a
ct − 2.04
− 1.61
− 1.62 − 2.89 − 3.91
− 12.02
− 16.59
− 18.00
− 3.04
1.90 − 4.93
− 2.62
B11th
e − 1.04
− 0.8045
− 0.7985
− 1.66 − 2.23
− 13.46
− 18.54
− 20.11
− 2.31
2.09 − 6.01
− 2.19
B12a
ct − 3.47
− 3.35
− 3.35 − 4.49 − 5.28
− 27.52
− 38.00
− 41.23
− 4.53
− 4.47
− 5.00
− 3.04
B12th
e − 1.52
− 1.01
− 0.9625
− 2.49 − 5.11
− 36.68
− 50.59
− 54.88
− 2.21
− 2.15
− 1.96
− 0.9048
R13a
ct 0.1709
0.1722
0.1695 0.2060 0.1300
5.52 7.59 8.24 0.2327
0.2276
0.3071
0.1612
R13t
he 0.7718
0.5713
0.5736 1.48 1.90 13.39
18.43
20.00
1.11 1.09 1.31 0.7122
R10act,1
− 0.1595
− 0.1197
− 0.1182
− 0.2235
− 0.1977
− 6.04
− 8.32
− 9.02
− 0.2527
0.1160
0.1516
0.0783
R10the,1
− 1.33
− 0.8068
− 0.8456
− 2.58 − 4.06
− 37.04
− 51.09
− 55.43
− 6.20
0.5394
0.6529
0.3605
R10act,2
0.0811
0.0614
0.0619 0.1218 0.1047
2.32 3.20 3.47 0.1663
0.1611
0.2242
0.1106
R10the,2
− 0.9205
− 0.4741
− 0.5587
− 1.42 − 3.05
− 8.17
− 11.26
− 12.22
− 1.21
− 1.17
− 1.44
− 0.8123
I16act 1.74 1.24 1.24 2.41 3.67 29.17
40.29
43.71
3.09 2.98 3.13 1.77
I16the 0.9277
0.5861
0.6108 2.02 2.25 35.66
49.16
53.33
1.49 1.43 1.67 0.9063
RVt(m)
BPVt(
m) RVt(ma.
adj1) RVt(ma.
adj2) MRt(Par)
MRt(
GK) MRt(
RS) MRt(
YZ) RRt(Par)
RRt(GK)
RRt(RS)
RRt(YZ)
C14act,1
2.10 1.93 1.93 3.82 4.38 36.34
50.18
54.44
2.81 2.78 3.50 1.85
C14the,1
2.79 1.90 1.89 3.19 5.93 46.59
64.27
69.73
4.42 4.37 4.82 2.78
C14act,2
2.20 1.94 1.79 3.12 3.70 44.23
61.05
66.23
2.28 2.22 2.73 1.44
C14act,2
2.72 1.70 1.77 3.28 5.82 40.73
56.16
60.93
3.19 3.15 3.58 2.09
C15a
ct 2.04 1.62 1.63 6.90 4.39 58.6
8 81.18
88.07
0.1517
0.1472
4.46 0.1051
C15t
he 0.0848
0.0468
0.0429 0.1052 0.4337
44.18
60.22
65.29
0.1186
0.1148
47.19 0.0817
Notes. Table 7 reports the average Sharpe ratio ( ) for both actual and theoretical BRIC Eurobond prices. Sharpe
ratio estimates (SRt) are split into the three groups of risk estimators: realized volatility, monthly range, and realized
range estimates. All average values reported, are t-test significant in a 5% significance level.
Table 8.
Mean magnitude of jumps ( ).
RVt(m)
BPVt(
m) RVt(ma. adj1)
RVt(ma. adj2)
MRt(Par)
MRt(GK)
MRt(RS)
MRt(YZ)
RRt(Par)
RRt(GK)
RRt(RS)
RRt(YZ)
B14a
ct 5.53e − 4
6.09e − 4
7.60e − 4
7.60e − 4
1.59e − 4
3.21e − 5
2.18e − 5
2.05e − 5
1.26e − 4
1.24e − 4
5.62e − 5
5.78e − 4
B14t
he 0.0013
6.67e − 4
0.0023
0.0022
5.32e − 4
1.71e − 5
1.83e − 5
1.70e − 5
0.0017
0.0021
0.0018
0.0020
B11a
ct 1.48e − 4
1.80e − 4
2.21e − 4
2.18e − 4
4.41e − 5
3.31e − 5
2.41e − 5
2.11e − 5
6.51e − 5
6.87e − 4
3.02e − 4
2.59e − 4
B11t
he 0.0013
1.23e − 4
5.90e − 4
5.97e − 4
3.68e − 4
3.40e − 5
1.95e − 5
2.08e − 5
6.23e − 4
6.23e − 4
6.23e − 4
6.23e − 4
B12a
ct 2.07e − 4
2.86e − 4
1.93e − 4
1.88e − 4
6.62e − 4
2.60e − 5
1.83e − 5
1.99e − 5
7.60e − 5
8.72e − 5
7.27e − 5
1.33e − 4
B12t
he 0.0016
8.56e − 4
0.0021
0.0023
3.56e − 4
3.30e − 5
2.25e − 5
2.21e − 5
6.14e − 4
6.14e − 4
6.14e − 4
6.14e − 4
R13act
3.81e − 4
0.0013
7.87e − 4
7.84e − 4
0.0015
2.13e − 5
1.42e − 5
1.14e − 5
2.07e − 4
2.88e − 4
1.26e − 4
3.74e − 4
R13t
he 0.0016
4.03e − 4
0.0012
0.0011
8.35e − 4
9.54e − 6
9.60e − 6
9.18e − 6
0.0019
0.0020
0.0015
7.89e − 4
R10act,1
3.71e − 4
0.0013
0.0021
0.0021
7.86e − 4
3.20e − 5
2.21e − 5
2.09e − 5
0.0028
0.0036
0.0029
0.0052
R10the,1
6.42e − 4
2.72e − 4
7.74e − 4
6.48e − 4
1.88e − 4
2.51e − 5
1.57e − 5
1.45e − 5
3.03e − 4
0.0032
0.0028
0.0049
R10act,2
6.10e − 4
6.52e − 4
7.95e − 4
9.66e − 4
3.92e − 4
3.02e − 5
1.83e − 5
1.55e − 5
1.69e − 4
1.73e − 4
6.50e − 5
3.46e − 4
R10t
he,2 5.57e − 4
2.42e − 4
8.41e − 4
6.27e − 4
1.81e − 4
2.59e − 5
1.44e − 5
1.34e − 5
2.20e − 4
2.31e − 4
1.99e − 4
2.98e − 4
RVt(m)
BPVt(
m) RVt(ma. adj1)
RVt(ma. adj2)
MRt(Par)
MRt(GK)
MRt(RS)
MRt(YZ)
RRt(Par)
RRt(GK)
RRt(RS)
RRt(YZ)
I16act 6.60e − 4
5.09e − 4
9.27e − 4
9.27e − 4
1.70e − 4
3.52e − 5
3.28e − 5
3.01e − 5
9.98e − 4
1.44e − 4
6.64e − 5
5.61e − 4
I16th
e 0.0053
8.07e − 4
0.0042
0.0040
0.0011
4.86e − 5
3.42e − 5
3.13e − 5
0.0057
0.0060
0.0059
0.0064
C14act,1
9.85e − 4
3.97e − 4
6.40e − 4
6.39e − 4
3.25e − 4
2.27e − 5
2.36e − 5
2.08e − 5
5.37e − 4
8.21e − 4
6.36e − 4
6.33e − 4
C14t
he,1 4.33e − 4
6.50e − 4
7.03e − 4
6.88e − 4
1.79e − 4
1.07e − 5
1.05e − 5
1.02e − 5
1.89e − 4
2.43e − 4
2.13e − 4
2.79e − 4
C14act,2
0.0031
3.65e − 4
0.0018
0.0018
0.0015
4.62e − 5
3.22e − 5
2.93e − 5
0.0042
0.0043
0.0032
0.0024
C14act,2
2.95e − 4
5.53e − 4
8.95e − 4
7.95e − 4
1.68e − 4
1.40e − 5
1.32e − 5
1.25e − 5
1.95e − 4
1.95e − 4
1.43e − 4
3.57e − 4
C15act
0.0321
0.0020
0.0574
0.0609
0.0151
3.75e − 5
3.33e − 5
3.05e − 5
0.0047
0.0050
0.0043
0.0034
C15t
he 3.50e − 4
0.0021
0.0390
0.0455
2.34e − 4
2.56e − 5
2.39e − 5
2.10e − 5
9.16e − 5
1.01e − 4
1.22e − 4
2.72e − 4
Notes. Table 8 reports the mean magnitude of jumps ( ) for both actual and theoretical BRIC Eurobond
prices. estimates are split into three groups: realized volatility, monthly range, and realized range estimates. All
average values reported, are t-test significant in a 5% significance level.
Table 9.
Average ratio of magnitude of the jump component of risk to the magnitude of the continuous component ( ).
RVt(
m) BPVt(
m) RVt(ma. a
dj1) RVt(ma. a
dj2) MRt(P
ar) MRt(
GK) MRt(
RS) MRt(
YZ) RRt(P
ar) RRt(
GK) RRt(R
S) RRt(Y
Z)
B14act 0.4804
10.48
0.7474 0.7474 0.7007
1.43 1.24 1.29 1.08 1.14 1.33 0.5161
B14the 2.33 9.07 2.02 1.87 2.60 0.4452
0 0 2.56 4.01 6.27 1.36
B11act 1.52 5.04 0.8948 0.8788 0.6849
1.73 1.76 1.55 1.02 8.21 5.02 1.22
B11the 5.47 3.06 2.46 2.49 1.88 1.85 1.06 1.46 5.10 5.10 5.10 5.10
B12act 1.84 7.51 0.4638 0.4261 0.7974
0.9587
0.9084
1.27 1.65 1.83 1.43 1.69
B12the 3.98 14.16
1.58 1.69 1.73 1.61 1.39 1.64 5.87 5.87 5.87 5.87
R13act 1.22 16.33
0.4443 0.4688 1.62 0.7531
0.6538
0.5243
0.6794
0.7149
1.11 0.5793
R13the 2.86 6.11 2.01 1.97 4.04 0.2249
0 0 6.18 6.43 4.62 1.63
R10a
ct,1 0.8622
22.25
1.03 1.06 2.25 1.83 1.61 1.72 7.91 32.93
26.61
32.59
R10the,1
2.33 5.71 1.78 1.54 1.27 0.9934
0.7503
0.7557
2.01 12.02
14.94
17.3
R10a 1.42 15.5 0.8640 1.05 0.743 1.58 1.05 0.89 1.21 1.26 2.22 1.54
RVt(
m) BPVt(
m) RVt(ma. adj1)
RVt(ma. adj2)
MRt(Par)
MRt(
GK) MRt(
RS) MRt(
YZ) RRt(Par)
RRt(
GK) RRt(RS)
RRt(YZ)
ct,2 6 7 14
R10th
e,2 2.89 5.26 2.16 1.67 1.30 1.07 0.65
71 0.6652
2.07 2.17 1.66 2.60
I16act 1.22 7.38 1.14 1.14 0.8589
2.63 18.00
16.51
2.47 1.03 1.06 1.27
I16the 13.15
8.66 2.96 2.79 3.50 15.11
10.65
9.74 13.20
13.79
10.45
14.34
C14a
ct,1 1.69 5.63 1.23 1.23 1.35 0.87
54 1.91 1.68 1.21 1.36 0.85
20 1.58
C14the,1
2.12 8.71 0.7765 0.7739 0.8771
0.2845
0 0 1.18 1.38 1.73 0.6720
C14a
ct,2 8.02 5.32 3.64 3.61 4.50 10.7
8 7.51 6.85 14.7
0 15.27
11.28
7.21
C14a
ct,2 0.7341
7.95 1.09 0.9977 1.21 0.3994
0 0 1.16 1.16 1.09 0.5176
C15act 53.26
16.06
2.06 2.18 32.92 2.93 10.76
9.84 4.00 4.16 2.73 3.65
C15the 0.8222
17.66
1.25 1.46 0.9940
1.02 1.87 1.64 0.3327
0.3667
0 0.8876
Notes. Table 9 reports the average ratio of magnitude of the jump component of risk to the magnitude of the
continuous component ( ) for both actual and theoretical BRIC Eurobond prices. estimates are split into three
groups: realized volatility, monthly range, and realized range estimates. All average values reported, are t-test
significant in a 5% significance level.
Table 10.
Average frequency of jump occurrence ( ).
RVt(
m) BPVt(
m) RVt(ma. a
dj1) RVt(ma. a
dj2) MRt(P
ar) MRt(
GK) MRt(
RS) MRt(
YZ) RRt(P
ar) RRt(
GK) RRt(R
S) RRt(Y
Z)
B14act 0.4167
0.9167⁎
0.5000⁎
0.5000⁎
0.3333
0.2222
0.1667
0.1389
0.2778
0.3056
0.2222
0.4167
B14the 0.3611
1.00⁎ 0.5556⁎
0.5556⁎
0.2500
0.0278
0.0278
0.0278
0.1389
0.1111
0.0833
0.2778
B11act 0.3889
0.7778⁎
0.8056⁎
0.8056⁎
0.4444
0.4167
0.3333
0.3333
0.4167
0.6667⁎
0.2500
0.7778⁎
B11the 0.2500
0.8889⁎
0.5000⁎
0.5000⁎
0.3611
0.2500
0.2500
0.1944
0.4722
0.4722
0.4722
0.4722
B12act 0.3611
0.8611⁎
0.6111⁎
0.6389⁎
0.3611
0.5000⁎
0.3333
0.2500
0.6389⁎
0.5278⁎
0.4167
0.8889⁎
B12the 0.2500
0.8611⁎
0.5556⁎
0.5556⁎
0.2778
0.2778
0.2222
0.1944
0.5000⁎
0.5000⁎
0.5000⁎
0.5000⁎
R13act 0.3333
0.9444⁎
0.4444 0.4722 0.6111⁎
0.3333
0.1667
0.1667
0.3056
0.2222
0.1944
0.6944⁎
R13the 0.2500
1.00⁎
0.6389⁎
0.6667⁎
0.1667
0.0833
0.0555
0.0555
0.1111
0.1111
0.1111
0.6111⁎
RVt(
m) BPVt(
m) RVt(ma. adj1)
RVt(ma. adj2)
MRt(Par)
MRt(
GK) MRt(
RS) MRt(
YZ) RRt(Par)
RRt(
GK) RRt(RS)
RRt(YZ)
R10a
ct,1 0.4167
0.8056⁎
0.5833⁎
0.5833⁎
0.5278⁎
0.3889
0.3333
0.3056
0.4167
0.8889⁎
0.8889⁎
0.9167⁎
R10the,1
0.3056
0.8889⁎
0.6944⁎
0.7222⁎
0.2222
0.2778
0.2222
0.1944
0.3056
0.9722⁎
0.9444⁎
0.9722⁎
R10a
ct,2 0.4444
0.6944⁎
0.5556⁎
0.5556⁎
0.5000⁎
0.5833⁎
0.5278⁎
0.5278⁎
0.7222⁎
0.7500⁎
0.5000⁎
0.9167⁎
R10th
e,2 0.2778
0.9167⁎
0.6944⁎
0.7222⁎
0.2222
0.2222
0.1944
0.1667
0.4444
0.4444
0.3333
0.6389⁎
I16act 0.2500
0.9167⁎
0.6111⁎
0.6111⁎
0.3056
0.0833
0.0556
0.0556
0.1111
0.1111
0.1111
0.3056
I16the 0.1944
0.9722⁎
0.5278⁎
0.5278⁎
0.3333
0.0278
0.0278
0.0278
0.1111
0.1111
0.0833
0.1667
C14act,1
0.3611
1.00⁎ 0.7222⁎
0.7222⁎
0.2500
0.0556
0.0278
0.0278
0.3333
0.2222
0.1667
0.7222⁎
C14the,1
0.1389
1.00⁎ 0.7222⁎
0.7500⁎
0.3333
0.0278
0.0278
0.0278
0.2222
0.1667
0.1111
0.6389⁎
C14a
ct,2 0.3056
0.9722⁎
0.7500⁎
0.7500⁎
0.2778
0.0278
0.0278
0.0278
0.1389
0.1389
0.1389
0.4167
C14act,2
0.2500
1.00⁎ 0.6667⁎
0.6944⁎
0.2500
0.0278
0.0278
0.0278
0.0833
0.0833
0.0556
0.3889
C15act 0.4444
0.9167⁎
0.7222⁎
0.7222⁎
0.3889
0.0833
0.0556
0.0556
0.1429
0.1429
0.1111
0.3571
C15the 0.1944
1.00⁎ 0.6667⁎
0.6667⁎
0.2778
0.0556
0.0278
0.0278
0.0833
0.0833
0.0833
0.1667
Notes. Table 10 reports the average frequency of jump occurrence ( ) for both actual and theoretical BRIC Eurobond
prices. estimates are split into three groups: realized volatility, monthly range and realized range estimates. All
average values reported, are t-test significant in a 5% significance level.
⁎
Indicates significance if the average frequency of jump occurrence ( ) is higher than 50%.
Table 11.
Theoretical vs actual prices.
RV 60% 50% 63% 68% 43%
MR 85% 35% 25% 28% 5%
RR 55% 68% 55% 60% 15%
Notes. Table 11 reports the average percentage of bonds for which a risk or jump estimate ( , , , and ) is
higher for theoretical prices rather than for actual prices.
Table 12.
Summarized results for BRIC countries.
J*
Brazil BPVt(m)/MRt(YZ) BPVt(m)/MRt(YZ) RRt(GK)/MRt(YZ) BPVt(m)/MRt(RS) BPVt(m)/MRt(YZ) 39%
Russia BPVt(m)/MRt(YZ) MRt(YZ)/BPVt(m) RRt(YZ)/MRt(YZ) BPVt(m)/MRt(RS) BPVt(m)/MRt(YZ) 49%
J*
India BPVt(m)/MRt(YZ) MRt(YZ)/BPVt(m) RRt(YZ)/MRt(YZ) MRt(RS)/RVt(ma. adj1) BPVt(m)/MRt(YZ) 25%
China BPVt(m)/MRt(YZ) MRt(YZ)/RRt(YZ) RVt(ma. adj1)/MRt(YZ) BPVt(m) / MRt(GK) BPVt(m)/MRt(YZ) 28%
Notes. Table 12 reports the estimators with the highest and (/) lowest values for the corresponding risk and jump
measures ( , , , and ). The last column provides the percentage of bonds (across estimators) for which
the frequency of occurrence of significant jumps is higher than 50% (J⁎).
Table 13.
Summarized results for groups of estimators.
RV China/Brazil China/Brazil China/Brazil China/Brazil Brazil/Russia
MR Brazil/India China/Russia India/Russia India/Russia Russia/Russia
RR China/Brazil China/Brazil India/Brazil Russia/China Russia/China
Notes. Table 13 reports the countries with the highest and (/) lowest risk and jump measures for each group of
estimators.
4.1. Unconditional distribution of return and volatility series
Many papers have examined the unconditional distribution of realized volatility (see Illueca &
Lafuente, 2006 and Wang, Wu, & Yang, 2008). Table 4 provides summary statistics for the
unconditional distribution of the realized volatilities. Volatility series for most of the BRIC
Eurobonds and for most of volatility estimators are skewed to the right (skewness higher than
zero) and leptokurtic (higher than three).
Table 5 deploys results for normality testing. The normality (CVM and QQ) tests do reject the
normality null hypothesis for most of the BRIC Eurobonds and across the board of estimators.
The critical value derived under independence for the CVM-test is 0.458 (5%); and for the
QQ-test is: 37.65 (5%). Most of volatilities (regardless either the group of estimators or the
country they belong to) are not normally distributed. The null hypothesis of normality is not
rejected for the MRt(GK), MRt(RS) and MRt(YZ) estimators. All results for the skewness and
kurtosis as well as for the normality testing of the unconditional distribution of volatilities are
consistent for both actual and theoretical prices.
4.2. Risk
Risk is measured via the mean of magnitude of risk ( ) (Table 6) and the mean of Sharpe
ratios ( ) (Table 7).
4.2.1. Average magnitude of risk ( ) The realized volatility (RV ) group of estimators has the highest mean magnitude of risk
series ( ) across all BRIC countries. The highest (lowest) mean of risk (Rt ) series ( ) comes
from the BPVt(m) (MRt(YZ)) estimator across BRIC Eurobonds, whereas the highest mean of
risk (Rt ) series ( ) comes from the Chinese Eurobonds across the board of estimators.
For the group of realized volatility estimators (RV) and the group of realized range-based
volatility estimators (RR), Chinese (Brazilian) Eurobonds have the highest (lowest) mean
magnitude of risk (Rt ) series ( ) among BRIC countries. For the group of monthly ranges
(MR ), Brazilian (Indian) Eurobonds have the highest (lowest) among BRIC countries.
The BPVt(m) (RVt(ma. adj2)) estimator has the highest (lowest) mean magnitude of risk (Rt ) series ( ) among the realized volatility estimators (RV), across all BRIC Eurobonds.
The MRt(YZ) (MRt(Par)) estimator has the highest (lowest) among the monthly ranges (MR),
across all BRIC Eurobonds. The RRt(RS) (RRt(YZ)) estimator has the highest (lowest) among
the realized range-based volatility estimators (RR), across all BRIC Eurobonds. Regarding
the performance of risk estimators, via the mean magnitude of risk (Rt ) series ( ), results are
consistent across all BRIC Eurobonds. All mean values of risk estimates for all bonds and
estimators are t-test statistically significant.
Moreover, the mean magnitude of risk (Rt ) series ( ) coming from theoretical prices
(theoretical-price risk) is higher than the mean risk coming from actual market prices (actual-
price risk). This result is evident in most of estimators and BRIC bonds. Across most of the
eurobonds, the higher the expiry period, the higher the mean magnitude of risk ( ) is.
4.2.2. Average Sharpe ratio ( ) The monthly range group of estimators (MR) has the highest mean of Sharpe ratios (SRt ) series ( ) across all BRIC countries. The highest (lowest) comes from
the MRt(YZ) (BPVt(m)) estimator across BRIC Eurobonds, whereas the highest
(lowest) comes from the Chinese (Brazilian) Eurobond across the board of estimators.
In specific, regarding Brazil, the RV realized volatility (MR monthly range) group of estimators
has the highest (lowest) mean of Sharpe ratios (SRt ) series ( ) among groups, and
the BPVt(m) (MRt(YZ)) estimator among individual estimators. Regarding Russia,
the MR monthly range (RV realized volatility) group of estimators has the highest
(lowest) among groups, and the MRt(YZ) (BPVt(m)) estimator among individual estimators.
Regarding India, the MR monthly range (RV realized volatility) group of estimators has the
highest (lowest) among groups, and the MRt(YZ) (BPVt(m)) estimator among individual
estimators. Regarding China, the MR monthly range (RR realized range) group of estimators
has the highest (lowest) among groups, and the MRt(YZ) (RRt(YZ)) estimator among
individual estimators.
For all three (RV, MR and RR) groups of estimators, Chinese (Brazilian) Eurobonds have the
highest (lowest) mean of Sharpe ratios (SRt ) series ( ) among BRIC countries.
The BPVt(m) (RVt(ma. adj2)) estimator has the highest (lowest) mean of Sharpe ratios (SRt) series
( ) among the realized volatility estimators, across all BRIC Eurobonds.
The MRt(Par) (MRt(YZ)) estimator has the highest (lowest) among the monthly ranges, across
all BRIC Eurobonds. The MRt(RS) (MRt(GK)) estimator has the highest (lowest) among the
realized range-based volatility estimators, across all BRIC Eurobonds.
Regarding the mean of Sharpe ratios (SRt ) series ( ), all estimators are consistent and
results change because of the informational content of BRIC bonds. All mean values of risk
estimates for all bonds and estimators are t-test statistically significant. Moreover, the mean of
Sharpe ratios (SRt ) series ( ) coming from theoretical prices (theoretical-price Sharpe
ratios) is higher than the mean Sharpe ratio coming from actual market prices (actual-price
Sharpe ratio). This result is evident in most of estimators and BRIC bonds. Across most of the
eurobonds, the higher the expiry period, the higher the mean magnitude of Sharpe ratios
(SRt ) series ( ) is.
4.3. Jumps
Eraker, Johannes, and Polson (2003) as well as more recently Atak and Kapetanios
(2013) provide results of significant average frequencies of occurrence (jump times) and
significant magnitudes of jumps (jump sizes). Barndorff-Nielsen and Shephard (2006) signify
the importance of jumps in asset prices compared to continuous sample paths,
whereas Todorov and Tauchen (2011) suggest that volatility is a pure jump process with
jumps of infinite variation. In the present paper, the significance of jumps is measured via the
mean magnitude of jumps ( ) (Table 8), the ratio of the mean magnitude of the jump
component of risk to the mean magnitude of the continuous component of risk ( )10 (Table 9)
and the average frequency of jump occurrence ( )11 (Table 10).
4.3.1. Average magnitude of jumps ( ) The RR realized range (MR monthly range) group of estimators has the highest (lowest)
mean magnitude of jumps ( ) series across all BRIC countries. The highest
(lowest) series comes from the RRt(YZ)(MRt(YZ)) estimator across BRIC Eurobonds,
whereas the highest (lowest) comes from the Indian (Brazilian) Eurobond across the
board of estimators.
In specific, regarding Brazil, the RR realized range (MR monthly range) group of estimators
has the highest (lowest) mean magnitude of jumps (JMt ) series ( ) among groups, and
the RRt(GK) (MRt(YZ)) estimator among individual estimators. Regarding Russia and India,
the RR realized range (RV realized volatility) group of estimators has the highest
(lowest) among groups, and the RRt(YZ) (MRt(YZ)) estimator among individual estimators.
Regarding China, the RV realized volatility (MR monthly range) group of estimators has the
highest (lowest) among groups, and the RVt(ma. adj1) (MRt(YZ)) estimator among individual
estimators.
For the group of realized volatility estimators, Indian (Brazilian) Eurobonds have the highest
(lowest) mean magnitude of jumps (JMt ) series ( ) among BRIC countries. For the group of
monthly ranges, Indian (Russian) Eurobonds have the highest (lowest) among BRIC
countries. For the group of realized range-based volatility estimators, Indian (Brazilian)
Eurobonds have the highest (lowest) among BRIC countries.
The RVt(ma. adj1) (BPVt(m)) estimator has the highest (lowest) mean magnitude of jumps (JMt ) series ( ) among the realized volatility estimators, across all BRIC Eurobonds.
The MRt(Par) (MRt(YZ)) estimator has the highest (lowest) among the monthly ranges,
across all BRIC Eurobonds. The RRt(YZ) (RRt(RS)) estimator has the highest
(lowest) among the realized range-based volatility estimators, across all BRIC Eurobonds.
Regarding the mean magnitude of jumps (JMt ) series ( ), all estimators are consistent and
results change because of the informational content of BRIC bonds. All mean values of jump
magnitude estimates for all bonds and estimators are t-test statistically significant. Moreover,
the mean magnitude of jumps (JMt ) series ( ) coming from theoretical prices (theoretical-
price Jump magnitudes) is higher than the mean magnitude of jumps coming from actual
market prices (actual-price Jump magnitudes). This result is evident in most of estimators and
BRIC bonds. Across most of the eurobonds, the higher the expiry period, the higher the mean
magnitude of jumps (JMt ) series ( ) is.
4.3.2. Average magnitude of jump component of risk relative to the magnitude of the
continuous component ( ) The RV realized volatility (MR monthly range) group of estimators has the highest (lowest)
mean (JRt ) series ( ) across all BRIC countries. The highest (lowest) comes from
the BPVt(m) (MRt(RS)) estimator across BRIC Eurobonds, whereas the highest
(lowest) comes from the Chinese (Brazilian) Eurobond across the board of estimators.
In specific, regarding Brazil and Russia, the RV realized volatility (MR monthly range) group
of estimators has the highest (lowest) mean of JRt series ( ) among groups, and
the BPVt(m) (MRt(RS)) estimator among individual estimators. Regarding India, the MR monthly
range (RV realized volatility) group of estimators has the highest (lowest) among groups,
and the MRt(RS) (RVt(ma. adj1)) estimator among individual estimators. Regarding China,
the RV realized volatility (MR monthly range) group of estimators has the highest
(lowest) among groups, and the BPVt(m) (MRt(GK)) estimator among individual estimators.
For the group of RV realized volatility estimators and the group of RR realized range
estimators, Chinese (Brazilian) Eurobonds have the highest mean of JRt series ( ) among
BRIC countries. For the group ofMR monthly ranges, Indian (Russian) Eurobonds have the
highest (lowest) among BRIC countries.
The BPVt(m) (RVt(ma. adj2)) estimator has the highest mean of JRt series ( ) among the
realized volatility estimators, across all BRIC Eurobonds. The MRt(Par) (MRt(RS)) estimator has
the highest (lowest) among the monthly ranges, across all BRIC Eurobonds.
The RRt(GK) (RRt(Par)) estimator has the highest (lowest) among the realized range-based
volatility estimators, across all BRIC Eurobonds.
Regarding the mean of JRt series ( ), all estimators are consistent and results change
because of the informational content of BRIC bonds. All mean ratios for all bonds and
estimators are t-test statistically significant. Moreover, the mean of JRt series ( ) coming
from theoretical prices (theoretical-price Jump ratio) is higher than the mean Jump ratio
coming from actual market prices (actual-price Jump ratio). This result is evident in most of
estimators and BRIC bonds. Across most of the eurobonds, the shorter the expiry period, the
higher the mean of JRt series ( ) is.
4.3.3. Average frequency of jump occurrences ( ) The RV realized volatility (MR monthly range) group of estimators has the highest (lowest)
average frequency of jump occurrence (Jt) series ( ) across all BRIC countries. The highest
(lowest) comes from the BPVt(m) (MRt(YZ)) estimator across BRIC Eurobonds, whereas the
highest (lowest) comes from the Brazilian (Russian) Eurobond across the board of
estimators.
For all BRIC countries, the RV realized volatility (MR monthly range) group of estimators has
the highest (lowest) average frequency of jump occurrence series among groups, and
the BPVt(m) (MRt(YZ)) estimator among individual estimators.
For the group of RV realized volatility estimators and the group of MR monthly ranges,
Brazilian (Russian) Eurobonds have the highest average frequency of jump occurrence
series among BRIC countries. For the group of RR realized range-based volatility
estimators, Russian (Chinese) Eurobonds have the highest among BRIC countries.
The BPVt(m) (RVt(m)) estimator has the highest (lowest) average frequency of jump
occurrence among the RV realized volatility estimators, across all BRIC Eurobonds.
The MRt(Par) (MRt(YZ)) estimator has the highest (lowest) among the MR monthly ranges,
across all BRIC Eurobonds. The RRt(YZ) (RRt(RS)) estimator has the highest (lowest) among
the RR realized range-based volatility estimators, across all BRIC Eurobonds.
Regarding the mean frequency of jump occurrence (Jt) series ( ), all estimators are consistent
and results change because of the informational content of BRIC bonds. All mean values of
jump magnitude estimates for all bonds and estimators are t-test statistically significant.
Moreover, the mean frequency of jump occurrence (Jt) series ( ) coming from theoretical
prices (theoretical-price Jump frequency) is higher than the mean frequency of jumps coming
from actual market prices (actual-price Jump frequencies). This result is evident in most of
estimators and BRIC bonds. Across most of the eurobonds, the higher the expiry period, the
higher the mean frequency of jump occurrence (Jt) series ( ) is.
Regarding the frequency of jump occurrence, Russian (Indian) Eurobonds have the highest
(lowest) number of estimators for which the is significant12. The RV realized volatility
(MR monthly range) group of estimators has the highest (lowest) number of Eurobonds for
which the is significant. The BPVt(m)(MRt(YZ)) estimator has the highest (lowest) number of
Eurobonds for which the is significant.
5. Conclusions
Concluding remarks concern results across all significance-measures: two risk significance-
measures ( , and ) and three jump significance-measures ( , and ). The overall
significance is evident when most of the significance measures are significant. The
significance of each either risk- or jump-measure is indicated as reported in the empirical
findings section. Firstly, findings are consistent as far as there are not many differences
between the group of the two risk measures and the group of the three jump measures.
Moreover, there are not many differences among the two risk measures and also among the
three jump measures. Moreover, all risk and jump measures from theoretical prices are higher
than those from actual prices, across bonds and estimators. Across most of the eurobonds
and measures, the higher the expiry period, the higher is the significance of risk and jumps.
This result is consistent with bond theory. The Chinese Eurobonds are the most significant,
across the board of estimators. Among BRIC Eurobonds, theC15the Eurobond is the most
significant.
The RV realized volatility group of estimators and the MR monthly range group of estimators
are the most significant (in terms of both risk and jumps) across all BRIC countries. The most
significant estimators areBPVt(m) bipower variation (high in risk and jumps)
and MRt(YZ) monthly Yang & Zhang range (low in risk and jumps) across BRIC Eurobonds. All
risk and jump significance-measures are consistent across the boards of estimators and BRIC
Eurobonds.
For all BRIC countries, the RV realized volatility (MR monthly range) group of estimators
retrieves the highest (lowest) estimates of risk and jumps. The bipower variation (monthly
Yang & Zhang range) estimator produce the highest (lowest) estimates of risk and jumps.
For the RV realized volatility group and the RR realized range group of estimators, the
Chinese (Brazilian) Eurobonds have the highest (lowest) estimates of risk and jumps among
all BRIC Eurobonds (countries). For the group of MR monthly ranges, Brazilian (Russian)
Eurobonds (Brazil) have the highest (lowest) estimates of risk and jumps among all BRIC
Eurobonds (countries). Risk and jump estimates are higher (lower) for theoretical prices
rather than actual prices for the RV realized volatility (MR monthly range) group of estimators.
The present paper suggests that theoretical prices are better to be used instead of the actual.
This empirical implication may trigger research on incorporating the theoretical pricing of
Eurobonds into modeling, forecasting and investing Eurobonds. As BRICs (Brazil, Russia,
India and China) become much larger force in the world economy, the accurate measurement
and the properties of BRIC Eurobonds risk will become more important in the international
financial markets and academia. The direct implications concern pricing structured products,
fund management, the predictability of risk, and international asset allocation.
Acknowledgment
The authors would like to thank the Editors Hamid Beladi and Carl R. Chen as well as two
anonymous referees for their valuable suggestions that improved the manuscript.
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Corresponding author. Tel.: + 44 1522 835634.
1
Tel.: + 44 1522 835612.
2
This acronym refers to the association of Brazil, Russia, India and China that first convened in
June 2009 ostensibly in response to the fallout of the 2008–09 financial crisis. In reality, it was
formed to offer an alternative framework of global governance anchored by the leading
emerging economies (see,Goodliffe & Sberro, 2012).
3
Eurobonds are issued offshore, in a currency different from that of the market where the bond
is arranged. The increased role of Eurobonds was signified by Henderson, Jegadeesh, and
Weisbach (2006), among others.
4
The recent financial crisis period had an international impact from February 2007 up to
February 2010. This is the target period of the present study.
5
Zhang, Mykland, and Ait-Sahalia (2005) showed that, in the presence of jumps, two-scales
realized volatility (TSRV) estimates the integrated variance plus the sum of squared intraday
jumps. A more recent analytical study is Zhang (2011).
6
Jacquier and Okou (2014) is a recent study in jumps in monthly realized volatility series.
7
More recent studies on the empirical applications, the properties of this detection scheme, the
properties of jumps series are Ait-Sahalia and Jacod (2011), Ait-Sahalia et al. (2012), and Ait-
Sahalia, Fan, and Li (2013).
8
These ratings were published on 12/31/2010. The Moody's ratings start from BB − (the worst
rating) to the A+(best rating) in the following order: BB −, BB, BB +, BBB −, BBB, BBB +, A −,
A, A +.
9
The asymptotic properties for their jump detection scheme were provided by Barndorff-
Nielsen and Shephard (2006) and Andersen et al. (2010) as well.
10
Significance is indicated if the mean magnitude of the jump component of risk relative to the
magnitude of the continuous component ( ) is higher than 1.
11
Significance is indicated if the average frequency of occurrence of jumps ( ) is higher than
50%.
12
Significance is indicated if the average frequency of jump occurrences ( ) is higher than 50%. Copyright © 2015 Published by Elsevier Inc.
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